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Full text of "Mathematical And Physical Papers - Iii"

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150       AN EXAMINATION  OF THE POSSIBLE  EFFECT OF THE
itself, because a ray of heat has only to traverse a length of air comparable with three or four diameters of the tube before it is absorbed by the tube. Hence we may conceive a small periodic flux of heat as taking place across the inner surface of the tube. Now it follows from the mathematical theory of heat, that when a periodic flux of heat takes place at the surface of a solid, the corresponding variation in the temperature of the solid near the surface is very small if the period be very small. If we suppose the flux expressed by the sine or cosine of an angle proportional to the time, the expression for the fluctuation of temperature will involve in its coefficient the square root of the period. In the present case, the period with which we have to deal is that of a sonorous vibration, a time which must be regarded as extremely small in questions relating to the conduction of heat. Hence, if r be the period of vibration, the fluctuation of temperature of the tube will be a small quantity of the order VT compared with the flux of heat. Now if H, h be the interior and exterior conductivities, v a normal to the inner surface of the tube, drawn from the tube inwards, & the excess of temperature of the tube above the mean temperature 00; and if we suppose the surface to be plane, and to radiate into an infinitely extended medium at a temperature #0 -J- 6, where 6 is supposed to be constant as regards space, but to be a periodic function of the time, we must have at the inner surface of the tube
Now, according to what has been already remarked, & is a small quantity of the order *Jr compared with dff'/dv, and it follows from the above equation, that dO1 ' jdv is comparable with 0, and therefore & is a small quantity of the order ^/r compared with 6. Hence, even in the case above supposed, the fluctuation of temperature of the tube at the surface would be very small. But in the actual case, the tube radiates, not into an infinite medium, but merely across the air contained within it, beyond which is situated the opposite face of the tube, at a temperature equal to the first face; and therefore the fluctuation of temperature of the inner surface of the tube will be far smaller than in the case supposed above, so as to be altogether insensible. Hence the air radiates within an envelope at a temperature #, so